Martensite Starting Temperature-Data Visualization & Data Exploration

Author

Hooman Sabarou-Cross_Validation Group_Fall 2024

Introduction to the Dataset (Martensite Starting Temperature)

This dataset contains the chemical compositions of various steels along with their Martensite starting temperatures (Ms), aiming to develop a model that predicts Ms based on steel chemistry.

Martensite is a phase where steel becomes extremely strong and can withstand high stresses, making it essential for industries like automotive to enhance crash safety. Ms is the critical temperature where steel’s internal structure changes into Martensite, and it varies with chemical composition. Currently, determining Ms requires lengthy tests using Thermogravimetry (TGA), which are prone to errors and often need repeating. A predictive model would save significant time and effort by eliminating the need for these tests.

The head of the dataset

     C   Mn   Si   Cr   Ni   Mo    V Co   Al W   Cu Nb Ti B     N     Ms
1 0.19 1.25 0.23 0.07 0.66 0.54 0.00  0 0.00 0 0.00  0  0 0 0.000 645.00
2 1.00 0.45 0.00 1.52 3.33 0.00 0.00  0 0.00 0 0.00  0  0 0 0.000 359.00
3 0.26 0.58 0.49 1.65 0.18 0.84 0.38  0 0.00 0 0.07  0  0 0 0.000 653.00
4 0.06 1.96 0.32 0.00 0.00 0.18 0.22  0 0.02 0 0.00  0  0 0 0.005 721.00
5 1.20 1.88 0.00 0.00 0.00 0.00 0.00  0 0.00 0 0.00  0  0 0 0.000 323.00
6 0.38 0.69 0.20 0.95 1.58 0.26 0.00  0 0.00 0 0.00  0  0 0 0.000 589.75

C, Mn, Si, Cr, and etc. are referred to chemical elements based on the Periodic Table. They are Carbon, Manganese, Silicon, Chromium, and so on.

Checking for missing values

[1] 0

There are no missing values.

Checking the Structure of the Dataset

'data.frame':   1543 obs. of  16 variables:
 $ C : num  0.19 1 0.26 0.06 1.2 0.38 0.43 0.004 0 2 ...
 $ Mn: num  1.25 0.45 0.58 1.96 1.88 0.69 0.83 0.03 0 0.65 ...
 $ Si: num  0.23 0 0.49 0.32 0 0.2 1.55 0.075 0 0.3 ...
 $ Cr: num  0.07 1.52 1.65 0 0 ...
 $ Ni: num  0.66 3.33 0.18 0 0 ...
 $ Mo: num  0.54 0 0.84 0.18 0 0.26 0.4 0 0 0.5 ...
 $ V : num  0 0 0.38 0.22 0 ...
 $ Co: num  0 0 0 0 0 0 0 0 0 0 ...
 $ Al: num  0 0 0 0.02 0 0 0 0 0 0.051 ...
 $ W : num  0 0 0 0 0 0 0 0 0 0 ...
 $ Cu: num  0 0 0.07 0 0 0 0 0 0 0.08 ...
 $ Nb: num  0 0 0 0 0 0 0 0 0 0 ...
 $ Ti: num  0 0 0 0 0 0 0 0 0 0 ...
 $ B : num  0 0 0 0 0 0 0 0 0 0 ...
 $ N : num  0 0 0 0.005 0 0 0 0.002 0 0 ...
 $ Ms: num  645 359 653 721 323 ...

The dataset has 1543 observations and 16 variables. Ms represent Martensite starting temperature and is the dependent variable.

The variables have continuous numeric content.

Summary statistics of the dataset

Ms is ranged between 150 to 800 degree Celsius. However, the most of the data are in 350 to 800 degree Celsius. The model may be designed for a limited range of temperature where we have more data (350 to 800C).

Carbon (C) has some zero contents which does not make any sense. Steel means Iron + Carbon. Iron without Carbon is not steel. C rows with zero content should be removed.

Scatter plot for one element and Martensite starting temperature

Carbon rows with zero content are recognized.

Histogram of Martensite starting temperature

Filtering the dataset for temperatures between 300 to 800 degree Celsius

Building GLM models

  1. Previous knowledge in Materials Science has been utilized to select important predictors. I am not sure if this is a correct approach.

Call:
glm(formula = Ms ~ C + Ni + Mn + Mo + Si + Cr + V + Co, family = gaussian, 
    data = filtered_dataset)

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  750.6469     3.1633 237.300  < 2e-16 ***
C           -260.1772     4.4241 -58.809  < 2e-16 ***
Ni           -12.4065     0.2859 -43.394  < 2e-16 ***
Mn           -25.5874     1.9657 -13.017  < 2e-16 ***
Mo            -7.7315     2.0623  -3.749 0.000184 ***
Si           -15.6230     2.9154  -5.359 9.69e-08 ***
Cr            -7.1898     0.5425 -13.254  < 2e-16 ***
V              8.3285     3.5131   2.371 0.017881 *  
Co             2.0491     0.9144   2.241 0.025184 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for gaussian family taken to be 1997.601)

    Null deviance: 12075332  on 1502  degrees of freedom
Residual deviance:  2984416  on 1494  degrees of freedom
AIC: 15699

Number of Fisher Scoring iterations: 2

At the first glance, V and Co present weak predictors. These two have to be removed.

Checking residuals

Q-Q plot shows deviations for extreme values.

Residual Distribution is skewed.

Identifying Outliers

         C   Mn      Si       Cr       Ni      Mo     V   Co    Al   W   Cu Nb
9    0.000 0.00 0.00000  0.53096 30.96549 0.00000 0.000 0.00 0.000 0.0 0.00  0
48   0.280 0.39 0.16000  2.35000  0.06000 0.06000 0.530 0.00 0.000 4.1 0.00  0
97   0.000 0.00 0.00000  0.00000 30.77361 1.36279 0.000 0.00 0.000 0.0 0.00  0
193  0.500 0.35 1.00000  0.11000  0.19000 0.50000 0.000 0.00 0.000 0.0 0.00  0
209  0.300 0.48 2.20000 10.50000  0.12000 1.00000 0.012 0.00 0.000 0.0 0.07  0
220  0.300 1.55 0.20000  0.00000  0.00000 0.28000 0.000 0.00 0.000 0.0 0.00  0
302  0.110 0.50 0.22000  0.00000  0.00000 0.56000 0.000 0.00 0.003 0.0 0.00  0
328  0.160 0.60 0.25000  0.20000  1.50000 0.05000 0.000 0.00 0.000 0.0 0.00  0
331  0.000 0.00 1.92824  0.00000 31.53985 0.00000 0.000 0.00 0.000 0.0 0.00  0
334  0.550 0.75 1.50000  0.70000  0.00000 0.00000 0.000 0.00 0.000 0.0 0.00  0
365  1.040 0.33 0.26000  1.53000  0.31000 0.01000 0.010 0.00 0.000 0.0 0.20  0
387  0.330 1.12 0.30000  0.11000  0.24000 0.04000 0.000 0.00 0.010 0.0 0.19  0
389  0.190 1.17 0.41000  0.06000  0.00000 0.00000 0.000 0.00 0.000 0.0 0.00  0
403  0.400 1.38 1.50000  0.00000  0.00000 0.80000 0.000 0.00 0.000 0.0 0.00  0
441  0.190 0.46 0.34000  7.83000  0.09000 2.02000 0.010 0.00 0.005 0.0 0.00  0
500  0.260 0.76 0.32000  1.08000  0.72000 1.25000 0.310 0.00 0.000 0.0 0.00  0
555  0.200 1.88 0.00000  0.00000  0.00000 0.00000 0.000 0.00 0.000 0.0 0.00  0
573  0.380 0.82 1.48000  0.72000  0.00000 0.77000 0.000 0.00 0.000 0.0 0.00  0
624  0.440 0.75 0.26000  1.70000  0.17000 0.08000 0.090 0.00 0.000 0.0 0.18  0
629  0.430 0.95 1.38000  1.06000  0.03000 0.10000 0.035 0.00 0.000 0.0 0.05  0
643  0.380 1.45 0.36000  0.00000  0.00000 0.76000 0.000 0.00 0.000 0.0 0.00  0
691  0.100 0.47 0.28000  1.32000  2.34000 0.00000 0.000 0.00 0.000 0.0 0.87  0
701  0.110 0.00 0.00000  0.00000  3.49000 0.00000 0.000 0.00 0.000 0.0 0.00  0
702  0.470 0.82 0.35000  1.20000  0.04000 0.00000 0.110 0.00 0.000 0.0 0.14  0
710  0.150 0.36 0.44000  2.24000  0.09000 0.85000 0.000 0.00 0.097 0.0 0.23  0
821  0.220 0.83 0.24000  0.54000  1.06000 0.51000 0.000 0.00 0.029 0.0 0.30  0
847  0.190 1.00 0.04000  0.62000  0.02000 0.00000 0.000 0.00 0.000 0.0 0.00  0
877  0.360 0.78 0.31000  0.00000  0.73000 0.49000 0.000 0.00 0.000 0.0 0.00  0
882  0.300 0.70 0.20000  0.00000  0.00000 0.00000 0.000 0.00 0.000 0.0 0.00  0
895  0.004 0.03 0.07500  0.00000 29.55000 0.00000 0.000 0.00 0.000 0.0 0.00  0
967  0.430 0.83 1.55000  0.91000  3.02000 0.40000 0.120 0.00 0.000 0.0 0.00  0
1035 0.400 1.47 0.37000  0.00000  0.00000 0.26000 0.000 0.00 0.000 0.0 0.00  0
1069 0.170 0.49 0.29000  0.18000  5.07000 0.24000 0.000 0.00 0.000 0.0 0.10  0
1091 0.330 0.74 0.23000  0.07000  3.47000 0.00000 0.000 0.00 0.000 0.0 0.00  0
1112 0.390 0.70 0.20000  1.05000  0.00000 0.00000 0.000 0.00 0.000 0.0 0.00  0
1144 0.120 0.46 0.35000  4.79000  0.20000 0.54000 0.000 0.00 0.000 0.0 0.00  0
1148 0.400 0.80 0.33000  0.00000  0.00000 0.79000 0.000 0.00 0.000 0.0 0.00  0
1166 0.400 0.75 0.27000  0.96000  0.13000 0.07000 0.060 0.00 0.000 0.0 0.20  0
1223 0.540 0.46 0.00000  0.00000  0.00000 0.00000 0.000 0.00 0.000 0.0 0.00  0
1225 0.470 0.40 1.06000  0.00000  0.00000 0.00000 0.000 0.00 0.000 0.0 0.00  0
1226 0.910 0.65 0.23000  0.60000  1.35000 0.00000 0.000 0.00 0.000 0.0 0.03  0
1242 0.870 1.78 0.29000  0.20000  0.15000 0.03000 0.000 0.00 0.000 0.0 0.00  0
1291 0.390 1.67 0.00000  0.00000  0.10000 0.00000 0.000 0.00 0.000 0.0 0.00  0
1302 0.360 1.50 0.20000  0.00000  0.00000 0.00000 0.000 0.00 0.000 0.0 0.00  0
1346 0.430 1.57 0.23000  0.12000  0.20000 0.07000 0.000 0.00 0.000 0.0 0.00  0
1398 0.300 0.40 0.30000  0.86000  3.20000 0.40000 0.000 0.00 0.000 0.0 0.00  0
1436 0.550 0.65 0.20000  0.00000  0.65000 0.00000 0.000 0.00 0.000 0.0 0.00  0
1492 0.760 0.25 0.35000  4.54000  0.00000 5.75000 2.050 0.86 0.000 6.6 0.00  0
        Ti B     N       Ms
9    0.000 0 0.000 218.1500
48   0.000 0 0.000 673.0000
97   0.000 0 0.000 199.1500
193  0.000 0 0.000 573.0000
209  0.000 0 0.000 528.0000
220  0.000 0 0.000 628.0000
302  0.000 0 0.002 775.0000
328  0.000 0 0.000 647.0000
331  0.000 0 0.000 171.1500
334  0.000 0 0.000 551.0000
365  0.000 0 0.000 518.0000
387  0.000 0 0.000 628.0000
389  0.000 0 0.000 700.0000
403  0.000 0 0.000 591.0000
441  0.000 0 0.013 623.0000
500  0.000 0 0.000 634.0000
555  0.000 0 0.000 666.0000
573  0.000 0 0.000 607.0000
624  0.000 0 0.000 573.0000
629  0.000 0 0.000 570.0000
643  0.000 0 0.000 601.0000
691  0.046 0 0.000 673.0000
701  0.000 0 0.000 722.0000
702  0.000 0 0.000 571.5000
710  0.010 0 0.000 698.0000
821  0.000 0 0.000 673.0000
847  0.000 0 0.000 705.0000
877  0.000 0 0.000 636.0000
882  0.000 0 0.000 666.0000
895  0.000 0 0.002 249.8167
967  0.000 0 0.000 568.0000
1035 0.000 0 0.000 596.0000
1069 0.000 0 0.000 641.0000
1091 0.000 0 0.000 583.0000
1112 0.000 0 0.000 606.0000
1144 0.000 0 0.000 736.0000
1148 0.000 0 0.000 603.0000
1166 0.035 0 0.010 634.0000
1223 0.000 0 0.000 596.0000
1225 0.000 0 0.000 594.2611
1226 0.000 0 0.000 422.0000
1242 0.000 0 0.000 422.0000
1291 0.000 0 0.000 618.0000
1302 0.000 0 0.000 603.0000
1346 0.000 0 0.000 572.0000
1398 0.000 0 0.000 598.0000
1436 0.000 0 0.000 548.0000
1492 0.000 0 0.000 473.0000

Remove rows where Carbon content is zero, update the model, and the summary of the model.

  1. The model has been updated after removing zero cells for Carbon content.

Call:
glm(formula = Ms ~ C + Ni + Mn + Mo + Si + Cr + V + Co, family = gaussian, 
    data = filtered_dataset_no_C0)

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  759.1787     3.0472 249.140  < 2e-16 ***
C           -266.4946     4.1713 -63.887  < 2e-16 ***
Ni           -14.0982     0.3195 -44.121  < 2e-16 ***
Mn           -29.0478     1.8747 -15.494  < 2e-16 ***
Mo            -8.0919     2.0278  -3.991 6.92e-05 ***
Si           -16.6972     2.7500  -6.072 1.61e-09 ***
Cr            -7.5038     0.5069 -14.804  < 2e-16 ***
V              8.3673     3.4113   2.453  0.01429 *  
Co             2.3044     0.8627   2.671  0.00764 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for gaussian family taken to be 1730.735)

    Null deviance: 11796080  on 1456  degrees of freedom
Residual deviance:  2506104  on 1448  degrees of freedom
AIC: 15010

Number of Fisher Scoring iterations: 2

Updating the model by removing Mo and Co and adding an interaction parameter between Carbon and Manganese.

  1. The model has been updated after removing weak predictors.

Call:
glm(formula = Ms ~ C * Mn + Ni + Si + Cr + V, family = gaussian, 
    data = filtered_dataset_no_C0)

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  749.7319     3.1524 237.829  < 2e-16 ***
C           -240.0692     5.6937 -42.164  < 2e-16 ***
Mn           -18.6538     2.3738  -7.858 7.54e-15 ***
Ni           -13.7303     0.3167 -43.358  < 2e-16 ***
Si           -13.2855     2.7810  -4.777 1.96e-06 ***
Cr            -8.3034     0.5069 -16.382  < 2e-16 ***
V              1.1567     3.1278   0.370    0.712    
C:Mn         -39.6809     6.0769  -6.530 9.09e-11 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for gaussian family taken to be 1701.745)

    Null deviance: 11796080  on 1456  degrees of freedom
Residual deviance:  2465829  on 1449  degrees of freedom
AIC: 14984

Number of Fisher Scoring iterations: 2

Update the model by removing V and add an interaction parameter between Carbon and Nickel.

  1. Further revision in the model by removing a weak parameter and adding a new interaction parameter

Call:
glm(formula = Ms ~ C * Mn + C * Ni + Si + Cr, family = gaussian, 
    data = filtered_dataset_no_C0)

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  751.5794     3.0913 243.129  < 2e-16 ***
C           -231.2782     5.4214 -42.660  < 2e-16 ***
Mn           -19.0683     2.3316  -8.178 6.23e-16 ***
Ni           -13.0691     0.3241 -40.326  < 2e-16 ***
Si           -15.1211     2.7398  -5.519 4.03e-08 ***
Cr            -8.5661     0.4913 -17.436  < 2e-16 ***
C:Mn         -41.8580     5.8686  -7.133 1.55e-12 ***
C:Ni         -13.8302     1.9610  -7.053 2.71e-12 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for gaussian family taken to be 1645.423)

    Null deviance: 11796080  on 1456  degrees of freedom
Residual deviance:  2384218  on 1449  degrees of freedom
AIC: 14935

Number of Fisher Scoring iterations: 2

Log Model

After reviewing Q-Q plot and its deviations in extreme values and a skewed residual distribution, a log model has been considered. It will be compared with the first model at the end.


Call:
glm(formula = log(Ms) ~ C * Mn + C * Ni + Si + Cr, family = gaussian, 
    data = filtered_dataset_no_C0)

Coefficients:
              Estimate Std. Error  t value Pr(>|t|)    
(Intercept)  6.6617940  0.0053830 1237.558  < 2e-16 ***
C           -0.4182892  0.0094406  -44.307  < 2e-16 ***
Mn          -0.0407894  0.0040602  -10.046  < 2e-16 ***
Ni          -0.0240965  0.0005644  -42.697  < 2e-16 ***
Si          -0.0245566  0.0047709   -5.147 3.01e-07 ***
Cr          -0.0152132  0.0008555  -17.783  < 2e-16 ***
C:Mn        -0.0698957  0.0102193   -6.840 1.17e-11 ***
C:Ni        -0.0248897  0.0034148   -7.289 5.12e-13 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for gaussian family taken to be 0.004989449)

    Null deviance: 37.6380  on 1456  degrees of freedom
Residual deviance:  7.2297  on 1449  degrees of freedom
AIC: -3578

Number of Fisher Scoring iterations: 2

Performing ANOVA for the updated model.

Loading required package: ggpubr

Performing ANOVA for the Log Model

Checking leverage and influential points for both models

Investigate the Influential Points

         C    Mn   Si    Cr   Ni   Mo    V Co Al W   Cu Nb Ti B     N     Ms
1145 0.022 10.24 0.00  8.19 0.00 0.00 0.00  0  0 0 0.00  0  0 0 0.206 327.15
718  2.250  0.00 0.00 11.50 0.00 0.80 0.20  0  0 0 0.00  0  0 0 0.000 422.00
719  2.080  0.39 0.28 11.48 0.31 0.02 0.04  0  0 0 0.15  0  0 0 0.000 457.00

By reviewing Carbon and Chromium contents, it can be decided that these three points are not representing routine chemistry for steels. They are unusual, and they can be removed.

Remove Influential Points and Updating Models


Call:
glm(formula = Ms ~ C * Mn + C * Ni + Si + Cr, family = gaussian, 
    data = filtered_dataset_no_influential)

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  753.7295     3.2150 234.445  < 2e-16 ***
C           -242.2463     5.5389 -43.735  < 2e-16 ***
Mn           -16.9666     2.6015  -6.522 9.58e-11 ***
Ni           -13.2149     0.3142 -42.053  < 2e-16 ***
Si           -15.0063     2.5997  -5.772 9.55e-09 ***
Cr            -9.1535     0.4711 -19.429  < 2e-16 ***
C:Mn         -40.5155     5.9696  -6.787 1.67e-11 ***
C:Ni         -12.4910     1.8647  -6.699 3.01e-11 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for gaussian family taken to be 1481.353)

    Null deviance: 11670630  on 1453  degrees of freedom
Residual deviance:  2142037  on 1446  degrees of freedom
AIC: 14751

Number of Fisher Scoring iterations: 2

Call:
glm(formula = log(Ms) ~ C * Mn + C * Ni + Si + Cr, family = gaussian, 
    data = filtered_dataset_no_influential)

Coefficients:
              Estimate Std. Error  t value Pr(>|t|)    
(Intercept)  6.6636794  0.0055789 1194.433  < 2e-16 ***
C           -0.4350978  0.0096117  -45.267  < 2e-16 ***
Mn          -0.0349821  0.0045145   -7.749 1.74e-14 ***
Ni          -0.0242615  0.0005453  -44.491  < 2e-16 ***
Si          -0.0243802  0.0045113   -5.404 7.60e-08 ***
Cr          -0.0161918  0.0008175  -19.806  < 2e-16 ***
C:Mn        -0.0707458  0.0103590   -6.829 1.25e-11 ***
C:Ni        -0.0225066  0.0032359   -6.955 5.31e-12 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for gaussian family taken to be 0.004460798)

    Null deviance: 37.1057  on 1453  degrees of freedom
Residual deviance:  6.4503  on 1446  degrees of freedom
AIC: -3733.4

Number of Fisher Scoring iterations: 2

Investigate Points Found Out by Cook’s Distances

        C   Mn   Si    Cr   Ni   Mo    V Co Al    W   Cu Nb Ti B N  Ms
1199 1.36 1.84 1.14  0.15 1.81 1.41 0.00  0  0 0.00 0.00  0  0 0 0 439
1259 2.08 0.39 0.28 11.48 0.31 0.02 0.04  0  0 0.00 0.15  0  0 0 0 361
1122 1.56 0.37 0.20 12.46 0.26 0.54 0.65  0  0 0.28 0.10  0  0 0 0 458

These observation are valid and will be kept in the dataset.

Apply the Residuals Check to the Filtered Dataset

Counting Outliers

        outlier    n
1 Not Suspected 1429
2     Suspected   25

Plotting Carbon Content with Martensite Starting Temperature

Performing ANOVA for Both Models After Removing Influential Points

Check Multicollinearity

First Model

       C       Mn       Ni       Si       Cr     C:Mn     C:Ni 
2.392626 2.309298 1.417059 1.067519 1.073750 2.869351 1.194094

Log Model

       C       Mn       Ni       Si       Cr     C:Mn     C:Ni 
2.392626 2.309298 1.417059 1.067519 1.073750 2.869351 1.194094

There is no concerning level of multicollinearity for both models.

Remove Identified Outliers and Revise Both Models


Call:
glm(formula = Ms ~ C * Mn + C * Ni + Si + Cr, family = gaussian, 
    data = filtered_dataset_no_influential_no_outlier)

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  769.4121     2.6216 293.489  < 2e-16 ***
C           -286.7145     4.5296 -63.299  < 2e-16 ***
Mn           -16.4197     2.1971  -7.473 1.36e-13 ***
Ni           -14.0369     0.2361 -59.454  < 2e-16 ***
Si           -13.8905     1.8663  -7.443 1.70e-13 ***
Cr           -10.1289     0.3825 -26.478  < 2e-16 ***
C:Mn         -41.4509     4.7323  -8.759  < 2e-16 ***
C:Ni          -8.3589     1.3578  -6.156 9.68e-10 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for gaussian family taken to be 760.2129)

    Null deviance: 10969684  on 1428  degrees of freedom
Residual deviance:  1080263  on 1421  degrees of freedom
AIC: 13545

Number of Fisher Scoring iterations: 2

Performing ANOVA for the First Model After Removing Outliers

Checking Cook’s Distance After Removing Outliers

Log Model After Removing Outliers


Call:
glm(formula = log(Ms) ~ C * Mn + C * Ni + Si + Cr, family = gaussian, 
    data = filtered_dataset_no_influential_no_outlier)

Coefficients:
              Estimate Std. Error  t value Pr(>|t|)    
(Intercept)  6.6889963  0.0043407 1540.990  < 2e-16 ***
C           -0.5102408  0.0074998  -68.034  < 2e-16 ***
Mn          -0.0319981  0.0036379   -8.796  < 2e-16 ***
Ni          -0.0255305  0.0003909  -65.309  < 2e-16 ***
Si          -0.0225859  0.0030902   -7.309 4.48e-13 ***
Cr          -0.0175028  0.0006334  -27.633  < 2e-16 ***
C:Mn        -0.0751043  0.0078355   -9.585  < 2e-16 ***
C:Ni        -0.0154270  0.0022482   -6.862 1.01e-11 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for gaussian family taken to be 0.002084123)

    Null deviance: 34.4909  on 1428  degrees of freedom
Residual deviance:  2.9615  on 1421  degrees of freedom
AIC: -4756.5

Number of Fisher Scoring iterations: 2

Performing ANOVA for The Log Model After Removing Outliers

Checking Cook’s Distance

Table 1 (Revised Dataset)

Variable Min Max Mean Median SD
Ms (Martensite Start Temp) 310.00 784.00 601.80 605.00 120.00
C (Carbon) 0.00 1.46 0.36 0.33 0.10
Mn (Manganese) 0.00 4.95 0.79 0.69 0.30
Ni (Nickel) 0.00 27.20 1.56 0.15 0.50
Si (Silicon) 0.00 3.80 0.35 0.26 0.20
Cr (Chromium) 0.00 16.20 1.04 0.52 0.70

Updated Summary for the Revised Dataset

       Ms              C                Mn               Ni        
 Min.   :310.0   Min.   :0.0016   Min.   :0.0000   Min.   : 0.000  
 1st Qu.:553.5   1st Qu.:0.1600   1st Qu.:0.4700   1st Qu.: 0.000  
 Median :605.0   Median :0.3300   Median :0.6900   Median : 0.150  
 Mean   :601.8   Mean   :0.3617   Mean   :0.7917   Mean   : 1.558  
 3rd Qu.:670.0   3rd Qu.:0.4400   3rd Qu.:0.9700   3rd Qu.: 1.580  
 Max.   :784.0   Max.   :1.4600   Max.   :4.9500   Max.   :27.200  
       Si               Cr        
 Min.   :0.0000   Min.   : 0.000  
 1st Qu.:0.2000   1st Qu.: 0.000  
 Median :0.2600   Median : 0.520  
 Mean   :0.3475   Mean   : 1.043  
 3rd Qu.:0.3400   3rd Qu.: 1.150  
 Max.   :3.8000   Max.   :16.200  
   vars    n   mean    sd median trimmed   mad min    max  range  skew kurtosis
Ms    1 1429 601.77 87.65 605.00  606.84 85.99 310 784.00 474.00 -0.49    -0.07
C     2 1429   0.36  0.26   0.33    0.33  0.22   0   1.46   1.46  1.27     1.79
Mn    3 1429   0.79  0.56   0.69    0.74  0.34   0   4.95   4.95  2.15    10.15
Ni    4 1429   1.56  3.82   0.15    0.70  0.22   0  27.20  27.20  4.37    20.23
Si    5 1429   0.35  0.40   0.26    0.26  0.10   0   3.80   3.80  3.26    15.13
Cr    6 1429   1.04  1.99   0.52    0.60  0.77   0  16.20  16.20  4.18    20.19
     se
Ms 2.32
C  0.01
Mn 0.01
Ni 0.10
Si 0.01
Cr 0.05

Histograms for Key Variables in the Revised Dataset

Table 2 Untransformed Model

Variables Mean ± SD Correlation Coefficient P-value
C 0.36 ± 0.1 -286.71 < 2e-16
Mn 0.79 ± 0.3 -16.42 1.36E-13
Ni 1.55 ± 0.5 -14.04 < 2e-16
Si 0.35 ± 0.2 -13.89 1.70E-13
Cr 1.04 ± 0.7 -10.13 < 2e-16
C:Mn N/A -41.45 < 2e-16
C:Ni N/A -8.36 9.68E-10

The model is:

\(Ms = 769.41 -286.71 C -16.42 Mn -14.04 Ni - 13.89 Si - 10.13Cr -41.45C:Mn - 8.36 C:Ni\)

Interpretation of the Coefficients

Each coefficient in this model tells us the marginal effect of that variable on Ms while keeping other variables constant.

  1. Intercept:

    • The intercept represents the predicted value of Ms when all predictors (C, Mn, Ni, Si, Cr) are at zero. This may not have a meaningful physical interpretation since, in practice, Ms is not typically defined when all elements are zero. However, it provides a baseline for the model.

  2. Main Effects:

    • Carbon (C): The coefficient for C represents how much Ms is expected to change for a one-unit increase in Carbon content, holding all other elements constant. A negative coefficient would indicate that higher Carbon content reduces the Martensite start temperature, which aligns with metallurgical theory, as carbon stabilizes the austenite phase, reducing Ms.

    • Manganese (Mn): The coefficient for Mn shows how Ms changes with an increase in Manganese. If this coefficient is negative, it suggests that Mn lowers Ms, likely due to its effect on stabilizing austenite.

    • Nickel (Ni): The coefficient for Ni indicates the effect of Nickel on Ms. Nickel also stabilizes austenite, so a negative coefficient would align with its known effect on lowering Ms.

    • Silicon (Si): The coefficient for Si represents how changes in Silicon content affect Ms. Silicon often raises Ms because it promotes ferrite formation.

    • Chromium (Cr): The coefficient for Cr reflects Chromium’s effect on Ms. Chromium typically lowers Ms as it also stabilizes the austenite phase, so we might expect a negative coefficient here.

  3. Interaction Parameters:

    • C and Mn

      Interaction: This term captures the combined effect of Carbon and Manganese on Ms. If significant, it suggests that the effect of Carbon on Ms depends on the level of Manganese, and vice versa. A negative interaction term would imply that as both C and Mn increase together, they have a compounded effect in reducing Ms more than either element alone.

    • C and Ni

      Interaction: Similarly, this term captures the interaction between Carbon and Nickel. If the coefficient is negative, it suggests that higher levels of both Carbon and Nickel together have an additional effect in lowering Ms, beyond their individual effects.

Table 2 Log-Transformed Model

Variables Mean ± SD Correlation Coefficient P-value
C 0.36 ± 0.1 -0.51 < 2e-16
Mn 0.79 ± 0.3 -0.032 < 2e-16
Ni 1.55 ± 0.5 -0.0255 < 2e-16
Si 0.35 ± 0.2 -0.0226 4.48E-13
Cr 1.04 ± 0.7 -0.0175 < 2e-16
C:Mn N/A -0.0751 < 2e-16
C:Ni N/A -0.0154 1.01E-11

The model is:

\(log(Ms) = -6.69 - 0.51C - 0.03 Mn - 0.03 Ni - 0.03 Si - 0.02Cr - 0.07 C:Mn - 0.01C:Ni\)

Interpretation of the Coefficients

Each coefficient in this model represents the proportional change in Ms for a unit change in each predictor, holding all other variables constant.

  1. Intercept:

    • The interceptrepres ents the expected value of log⁡(Ms) when all predictors are zero. Exponentiating this term gives the baseline value of Ms for a sample with zero values for C, Mn, Ni, Si, Cr. This value is hypothetical as these elements are rarely all zero in practice.

  2. Main Effects:

    • Carbon (C): The coefficient for C  \(\beta_C\) represents the proportional change in Ms for each additional unit of Carbon. Since the response is log-transformed, a unit increase in C multiplies Ms by \(e^{\beta_C}\) . If \(\beta_C\) is negative, it indicates that increasing Carbon reduces Ms, consistent with Carbon’s known effect of stabilizing austenite.

    • Manganese (Mn): The coefficient for Mn \(\beta_{Mn}\) shows the effect of Manganese on the log of Ms. A negative value indicates that Manganese decreases Ms in a multiplicative way, meaning each increase in Manganese content corresponds to a percentage decrease in Ms.

    • Nickel (Ni): The coefficient for Ni \(\beta_{Ni}\) indicates Nickel’s impact on the log of Ms. Like Manganese, a negative coefficient would imply that increasing Nickel decreases Ms proportionally. This aligns with Nickel’s effect as an austenite stabilizer.

    • Silicon (Si): The coefficient for Si \(\beta_{Si}\) captures Silicon’s impact on the log of Ms. Silicon often raises Ms by promoting ferrite formation, so a positive coefficient would imply that increasing Silicon raises Ms proportionally.

    • Chromium (Cr): The coefficient for Cr \(\beta_{Cr}\) represents Chromium’s effect on the log of Ms. Chromium generally lowers Ms due to its role in stabilizing austenite. A negative value here would confirm this behavior.

  3. Interaction Parameters:

    • Interaction (\(\beta_{C:Mn}\)): This term captures the combined effect of Carbon and Manganese on the log of Ms. If significant, it suggests that the effect of Carbon on Ms depends on the level of Manganese, and vice versa. A negative interaction term means that as both C and Mn increase, they together reduce Ms more than each would individually.

    • Interaction (\(\beta_{C:Ni}\)): Similarly, this term captures the interaction between Carbon and Nickel. A negative coefficient here would imply that higher levels of both Carbon and Nickel together have an added effect in reducing Ms beyond their individual effects.

Cross-Validation

5-Fold

Cross-Validation for the First Model

5-Fold

[1] 774.1075 772.0337

10-Fold

[1] 779.8235 778.5136

Interpretation

  1. Consistency: Both 5-fold and 10-fold cross-validations yield fairly similar results, which indicates stability in the model’s predictive performance across different validation approaches.

  2. Slight Increase with 10-fold: The error slightly increased with 10-fold cross-validation. This could be due to the smaller subsets in each fold, as 10-fold divides the data into smaller groups compared to 5-fold, potentially revealing more variability or minor overfitting effects.

  3. Implications for Model Evaluation: These results suggest that while the untransformed model’s performance is stable, there is no significant improvement in prediction error when using more folds. However, if computational efficiency or consistency is a priority, 5-fold might be a suitable choice.

Cross-Validation for the Second Model (Log Model)

5-Fold

[1] 0.002146943 0.002138416

10-Fold

[1] 0.002144025 0.002140193

Interpretation

  1. Stability Across Folds: Both the 5-fold and 10-fold cross-validation results for the log-transformed model are extremely close, with very little variation between the fold types. This suggests that the log-transformed model is highly stable and performs consistently across different subsets of the data.

  2. Comparison with Untransformed Model:

    • The cross-validation errors for the log-transformed model (~0.0021) are significantly lower than those of the untransformed model (~774 to ~780). This indicates that the log-transformed model likely fits the data better and generalizes more effectively.

    • The log-transformed model’s lower error suggests it may be less sensitive to outliers or non-normality, providing a more reliable predictive performance.

  3. Selection of the Log-Transformed Model: Given the much lower cross-validation errors and consistency across folds, the log-transformed model is clearly outperforming the untransformed model. This makes it the preferable choice for accurate and robust predictions of the Martensite start temperature.

The Leave-One-Out Cross-Validation (LOOCV)

The Untransformed Model

[1] 772.7368 772.7308

The Log-Transformed Model

[1] 0.002125407 0.002125388

Model Comparison

  • Predictive Accuracy: The log-transformed model performs better in terms of LOOCV error, suggesting it is more reliable for prediction. This result aligns with the findings from the earlier steps, where the log-transformed model consistently showed lower residual deviance and AIC.

  • Practical Use: If the purpose of a model is interpretability or making predictions on the original scale of Ms, the untransformed model may still be relevant despite the higher LOOCV error. However, for optimal prediction accuracy, the log-transformed model is superior based on these results.

Conclusion

The log-transformed model shows a more stable and lower prediction error with LOOCV, supporting its choice as the better model in terms of predictive performance.